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Sep 22, 2013
09/13

by
A. Mohammad-Djafari

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The main object of this paper is to present some general concepts of Bayesian inference and more specifically the estimation of the hyperparameters in inverse problems. We consider a general linear situation where we are given some data $\yb$ related to the unknown parameters $\xb$ by $\yb=\Ab \xb+\nb$ and where we can assign the probability laws $p(\xb|\thetab)$, $p(\yb|\xb,\betab)$, $p(\betab)$ and $p(\thetab)$. The main discussion is then how to infer $\xb$, $\thetab$ and $\betab$ either...

Source: http://arxiv.org/abs/physics/0111123v1

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Sep 22, 2013
09/13

by
A. Mohammad-Djafari

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To handle with inverse problems, two probabilistic approaches have been proposed: the maximum entropy on the mean (MEM) and the Bayesian estimation (BAYES). The main object of this presentation is to compare these two approaches which are in fact two different inference procedures to define the solution of an inverse problem as the optimizer of a compound criterion. Keywords: Inverse problems, Maximum Entropy on the Mean, Bayesian inference, Convex analysis.

Source: http://arxiv.org/abs/physics/0111122v1

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657

Sep 22, 2013
09/13

by
A. Mohammad-Djafari

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The classical Maximum Entropy (ME) problem consists of determining a probability distribution function (pdf) from a finite set of expectations of known functions. The solution depends on $N+1$ Lagrange multipliers which are determined by solving the set of nonlinear equations formed by the $N$ data constraints and the normalization constraint. In this short communication we give three Matlab programs to calculate these Lagrange multipliers. The first considers the general case where the...

Source: http://arxiv.org/abs/physics/0111126v1

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Sep 22, 2013
09/13

by
A. Mohammad-Djafari

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The Bayesian approach has proved to be a coherent approach to handle ill posed Inverse problems. However, the Bayesian calculations need either an optimization or an integral calculation. The maximum a posteriori (MAP) estimation requires the minimization of a compound criterion which, in general, has two parts: a data fitting part and a prior part. In many situations the criterion to be minimized becomes multimodal. The cost of the Simulated Annealing (SA) based techniques is in general huge...

Source: http://arxiv.org/abs/physics/0111121v1

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Sep 22, 2013
09/13

by
A. Mohammad-Djafari

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A complete solution for an inverse problem needs five main steps: choice of basis functions for discretization, determination of the order of the model, estimation of the hyperparameters, estimation of the solution, and finally, characterization of the proposed solution. Many works have been done for the three last steps. The first two have been neglected for a while, in part due to the complexity of the problem. However, in many inverse problems, particularly when the number of data is very...

Source: http://arxiv.org/abs/physics/0111020v1

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Sep 22, 2013
09/13

by
A. Mohammad-Djafari

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Image reconstruction in X ray tomography consists in determining an object from its projections. In many applications such as non destructive testing, we look for an image who has a constant value inside a region (default) and another constant value outside that region (homogeneous region surrounding the default). The image reconstruction problem becomes then the determination of the shape of that region. In this work we model the object (the default region) as a polygonal disc and propose a...

Source: http://arxiv.org/abs/physics/0111120v1

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Sep 22, 2013
09/13

by
A. Mohammad-Djafari

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The main object of this paper is to show how we can use classical probabilistic methods such as Maximum Entropy (ME), maximum likelihood (ML) and/or Bayesian (BAYES) approaches to do microscopic and macroscopic data fusion. Actually ME can be used to assign a probability law to an unknown quantity when we have macroscopic data (expectations) on it. ML can be used to estimate the parameters of a probability law when we have microscopic data (direct observation). BAYES can be used to update a...

Source: http://arxiv.org/abs/physics/0111118v1

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Sep 22, 2013
09/13

by
A. Mohammad-Djafari; Jérôme Idier

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In this paper we propose a new Bayesian estimation method to solve linear inverse problems in signal and image restoration and reconstruction problems which has the property to be scale invariant. In general, Bayesian estimators are {\em nonlinear} functions of the observed data. The only exception is the Gaussian case. When dealing with linear inverse problems the linearity is sometimes a too strong property, while {\em scale invariance} often remains a desirable property. As everybody knows...

Source: http://arxiv.org/abs/physics/0111125v1

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Sep 22, 2013
09/13

by
A. Mohammad-Djafari; Ken Sauer

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X-ray tomographic image reconstruction consists of determining an object function from its projections. In many applications such as non-destructive testing, we look for a fault region (air) in a homogeneous, known background (metal). The image reconstruction problem then becomes the determination of the shape of the default region. Two approaches can be used: modeling the image as a binary Markov random field and estimating the pixels of the image, or modeling the shape of the fault and...

Source: http://arxiv.org/abs/physics/0111117v1

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Sep 22, 2013
09/13

by
A. Mohammad-Djafari; H. G. Miller

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The problem of the determination of the charge density from limited information about the charge form factor is an ill-posed inverse problem. A Bayesian probabilistic approach to this problem which permits to take into account both errors and prior information about the solution is presented. We will show that many classical methods can be considered as special cases of the proposed approach. We address also the problem of the basis function choice for the discretization and the uncertainty of...

Source: http://arxiv.org/abs/physics/0111119v1

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Sep 22, 2013
09/13

by
Doriano-Boris Pougaza; A. Mohammad-Djafari; Jean-François Bercher

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In 1917 Johann Radon introduced the Radon transform which is used in 1963 by A. M. Cormack for application in the context of tomographic image reconstruction. He proposed to reconstruct the spatial variation of the material density of the body from X-Ray images (radiographies) for different directions. Independently G. N. Hounsfield derived an algorithm and built the first medical CT scanner. Basically the idea of the X-ray CT is to get an image of the interior structure of an object by...

Source: http://arxiv.org/abs/0812.1316v1

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Sep 23, 2013
09/13

by
H. Snoussi; G. Patanchon; J. F. Macias-Perez; A. Mohammad-Djafari; J. Delabrouille

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We present a technique for the blind separation of components in CMB data. The method uses a spectral EM algorithm which recovers simultaneously component templates, their emission law as a function of wavelength, and noise levels. We test the method on Planck HFI simulated observations featuring 3 astrophysical components.

Source: http://arxiv.org/abs/astro-ph/0109123v1

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Jun 29, 2018
06/18

by
V. Abrol; O. Absil; P. -A. Absil; S. Anthoine; P. Antoine; T. Arildsen; N. Bertin; F. Bleichrodt; J. Bobin; A. Bol; A. Bonnefoy; F. Caltagirone; V. Cambareri; C. Chenot; V. Crnojević; M. Daňková; K. Degraux; J. Eisert; J. M. Fadili; M. Gabrié; N. Gac; D. Giacobello; A. Gonzalez; C. A. Gomez Gonzalez; A. González; P. -Y. Gousenbourger; M. Græsbøll Christensen; R. Gribonval; S. Guérit; S. Huang; P. Irofti; L. Jacques; U. S. Kamilov; S. Kiticć; M. Kliesch; F. Krzakala; J. A. Lee; W. Liao; T. Lindstrøm Jensen; A. Manoel; H. Mansour; A. Mohammad-Djafari; A. Moshtaghpour; F. Ngolè; B. Pairet; M. Panić; G. Peyré; A. Pižurica; P. Rajmic; M. Roblin; I. Roth; A. K. Sao; P. Sharma; J. -L. Starck; E. W. Tramel; T. van Waterschoot; D. Vukobratovic; L. Wang; B. Wirth; G. Wunder; H. Zhang

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The third edition of the "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) took place in Aalborg, the 4th largest city in Denmark situated beautifully in the northern part of the country, from the 24th to 26th of August 2016. The workshop venue was at the Aalborg University campus. One implicit objective of this biennial workshop is to foster collaboration between international scientific teams by disseminating ideas through both...

Topics: Computer Vision and Pattern Recognition, Numerical Analysis, Mathematics, Optimization and Control,...

Source: http://arxiv.org/abs/1609.04167